X^2-4x-1=2k(x-5)

Question

Find the value of k and sove the equation

Answer 1

x²-4x-1=2k(x-5)=2kx-10k,

x²-4x-2kx=1-10k,

x²-2x(2+k)=1-10k,

x²-2x(2+k)+(2+k)²=

(2+k)²+1-10k=5-6k+k²=(5-k)(1-k).

(x-(2+k))²=(5-k)(1-k).

If k=1 or 5, x=k+2,

so x=3 or 7 for a rational solution.

CHECK

x=3, k=1:

9-12-1=2(-2)=-4 OK

x=7, k=5:

49-28-1=10(2)=20 OK